Vasylkevych, Sergiy and Marsden, Jerrold E. (2005) The Lie-Poisson Structure of the Euler Equations of an Ideal Fluid. Dynamics of Partial Differential Equations, 2 (4). pp. 281-300. ISSN 1548-159X. http://resolver.caltech.edu/CaltechAUTHORS:20100917-074331134
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This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in R^n (or a smooth compact n-manifold with boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphism group. This is interesting and nontrivial because in Eulerian representation, the time t maps need not be C^1 from the Sobolev class H^s to itself (where s > (n=2) + 1). The idea of how this diculty is overcome is to exploit the fact that one does have smoothness in the Lagrangian representation and then carefully perform a Lie-Poisson reduction procedure.
|Additional Information:||© 2005 International Press. Communicated by Tudor Ratiu, received August 23, 2005. The hardcopy and electronic editions of Dynamics of Partial Differential Equations are protected by the copyright of International Press.|
|Subject Keywords:||Euler equations, Poisson map, Lie-Poisson bracket, Lagrangian representation, Lie-Poisson reduction procedure.|
|Classification Code:||1991 MSC: Primary: 35; Secondary: 76.|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Ruth Sustaita|
|Deposited On:||17 Sep 2010 21:17|
|Last Modified:||26 Dec 2012 12:26|
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